Help With Targeting Players?

From my knowledge, normal scripts will not work inside PlayerGui. If you want a player to have control via GUI over something all players see (a server-side thing like your zombies) you'll have to use Remote Events and a Local Script. I'll give you my rendition of this, but promise me you'll learn from it.

This will be a normal script located in ServerScriptService (You can put it anywhere not related to a player, but I would put it in there):

local ZAttack = Instance.new("RemoteEvent", game.ReplicatedStorage)
ZAttack.Name = "ZAttack"
ZAttack.OnServerEvent:connect(function(player,name) --'player' is the client that calls the function, 'name' is what that player inputs into the function's argument(or was it called parameter? idk I don't have formal training.)
    local target = game.Players:FindFirstChild(name)
    if target == nil then return end --Switched out the check if the target exists
    for i,z in pairs(game.Workspace.Zombies:GetChildren()) do
        if z.Owner.Value ~= player.Name then --Your original check would've left you vulnerable too!
            z:MoveTo(target.Character.Torso) --Players don't have torsos but their characters do!
        end
    end
end)

And put this in a Local Script that will replace your current one:

script.Parent.MouseButton1Click:connect(function()
    game.ReplicatedStorage.ZAttack:FireServer(script.Parent.Parent.Name.Text) --That's the Argument/Parameter in the parenthesis that I mentioned before!
end)

tl;dr I did it for you, dissect my script(s) so you can do it too!